Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-41475x+3231250\)
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(homogenize, simplify) |
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\(y^2z=x^3-41475xz^2+3231250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-41475x+3231250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(65, 900)$ | $1.0799985232968632405643373432$ | $\infty$ |
| $(110, 0)$ | $0$ | $2$ |
| $(125, 0)$ | $0$ | $2$ |
Integral points
\( \left(-235, 0\right) \), \((35,\pm 1350)\), \((65,\pm 900)\), \( \left(110, 0\right) \), \( \left(125, 0\right) \), \((126,\pm 76)\), \((179,\pm 1242)\), \((225,\pm 2300)\), \((5525,\pm 410400)\)
Invariants
| Conductor: | $N$ | = | \( 82800 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $55532304000000$ | = | $2^{10} \cdot 3^{8} \cdot 5^{6} \cdot 23^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{676449508}{4761} \) | = | $2^{2} \cdot 3^{-2} \cdot 7^{3} \cdot 23^{-2} \cdot 79^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4701099389481512961537049856$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.46153781206957482802532406736$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9919362448497163$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8424087698022946$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0799985232968632405643373432$ |
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| Real period: | $\Omega$ | ≈ | $0.63168642590179707092285533210$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.4577632572849140724489731619 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.457763257 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.631686 \cdot 1.079999 \cdot 128}{4^2} \\ & \approx 5.457763257\end{aligned}$$
Modular invariants
Modular form 82800.2.a.l
For more coefficients, see the Downloads section to the right.
| Modular degree: | 262144 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $23$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 291 & 2210 \\ 910 & 551 \end{array}\right),\left(\begin{array}{rr} 2619 & 1100 \\ 2200 & 2749 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2757 & 4 \\ 2756 & 5 \end{array}\right),\left(\begin{array}{rr} 919 & 1100 \\ 2390 & 2199 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1103 & 0 \\ 0 & 2759 \end{array}\right),\left(\begin{array}{rr} 1381 & 1660 \\ 2210 & 561 \end{array}\right)$.
The torsion field $K:=\Q(E[2760])$ is a degree-$196977623040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2760\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $8$ | \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \) |
| $5$ | additive | $14$ | \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 82800bq
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 552e2, its twist by $60$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-10}, \sqrt{138})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{10})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-23})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | add | ord | ss | ord | ord | ord | split | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | - | 1 | 1,3 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | - | - | - | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.